Imaging Spectroscopy
Imaging spectroscopy is a digital sensing process in which a scene is optically sampled in two spatial dimensions and in one spectral dimension, producing a three dimensional data cube. Spectral images contain significantly larger amount of spectral information than color photography, typically comprising tens or hundreds of well defined narrow spectral channels for each individual pixel of the image. In other words each spatial pixel of the spectral image contains a spectral response of the respective point of the imaged surface. In comparison, a color image is comprised by three loosely defined Red-Green-Blue channels. Spectral images further distinguish themselves from color photography through the rigorously measured radiometric output: while color is perception based and has no absolute units, spectral radiance is a physical measure whose unit is [Wm−2 μm−1 sr−1]. As a consequence, spectral imaging is often used to determine the chemical and biological composition of objects, while avoiding the need for physical contact [1, 2]. The acquisition of 3D hyperspectral imaging data is difficult because of the two-dimensional nature of the imaging sensors.
In order to obtain the spectral data cube with a two dimensional sensor, most spectral imaging cameras use the following three elements:                1. An optical element, such as a lens: to focus the optical scene onto an imaging plane.        2. A dispersive element, such as a prism or diffraction grid: to spatially distribute spectral information from the imaging plane.        3. An imaging sensor, such as a CCD or a CMOS: to spatially sample the dispersed light.        
Due to the spectral dispersion, spatial resolution must be compromised. To overcome the resolution loss, many spectral imaging cameras employ some form of scanning:                Pushbroom, or line-scan, cameras capture a spatio-spectral slice of the datacube, using the two dimensions of the imaging sensor as 1D spectral and 1D spatial sampling. Pushbroom sensors require movement to reconstruct an entire hyperspectral image, they require spatial scanning.        Band sequential cameras capture a spectral slice of the spectral data cube, using either rotating or tunable spectral filters. They use the two dimensions of the imaging sensor as 2D spatial sampling and scan spectrally by sequentially applying different spectral filters in front of the imaging sensor, they thus require spectral scanning.        Interferometry-based spectral cameras sequentially capture interferograms of the data cube onto the image sensor. The interferograms are generated by tuning an interferometer, a process similar to spectral scanning.        
Frame, or snapshot, spectral cameras acquire the entire data cube without scanning, severely compromising resolution. State-of-the-art snapshot spectral cameras are either based on 2D diffraction grids or small filter banks, requiring a high resolution sensor and producing very low resolution spectral data cubes with respect to scanning spectral cameras [2].
Compressive Sensing
Classical signal processing dictates that in order to sample, then reconstruct a signal without information loss, the signal has to be sampled with at least twice the highest frequency it contains. This minimal sampling frequency is also called Nyquist frequency. If sampling is done under the Nyquist frequency, the reconstructed signal will show information loss and artefacts such as aliases. The large number of measurements required to reconstruct a full hyperspectral data cube under the Nyquist constraint is one of the main reasons why all non-compressive snapshot spectral cameras have low resolution.
Recent research in signal processing and advances in computational power have led to many improvements in signal acquisition bandwidth reduction through compressive sensing. At the core of compressive sensing lays the hypothesis that if the signal that needs to be acquired is sparse in some mathematical basis, the number of measurements required to reconstruct the signal are given by its sparsity and can thus be significantly smaller than the number of measurements dictated by the Nyquist frequency.
The effective reconstruction of the original signal using the compressive sensing mathematical models imposes certain requirements on the underlying signal sampling methodology, in particular uniformity and aperiodicity.
In areas of applications such as magnetic resonance imaging (MRI), compressive sensing has been successfully implemented in order to reduce the amount of required measurements by a factor of between 10 and 20 [4], or, with a corresponding compression ratio of between 1/10 and 1/20. For color images, the method has been suggested in [5], but the practical implementation has been limited by the low information gain and high computational cost with respect to classical demosaicing. In color photography, the required compression ratios is 1/3 as the images have a small quantity of spectral information.
Compressive sensing has also been applied to spectral imaging in multiple scientific experiments, showing compression ratios of between 1/4 and 1/16, [6][7]. These experiments, however, failed to produce the results needed for a practical snapshot spectral camera which requires compression ratios around 1/50 or more. The color imaging methods described in [5] cannot be directly applied to 2D spectral sampling-based imaging for a plurality of reasons:                Correct pixel-wise spectral reconstruction poses a series of new challenges such as optimal spectral distribution of filters, or higher order filter response cancellation.        Color filter arrays do not provide sufficient spectral separation        For hyperspectral image reconstruction, the algorithms need to reconstruct a significantly larger set of data from less than 5% of its size, as opposed to 33% subsampling rate of color imaging.        
Interferometric Spectral Filter Array
Due to the very stringent requirements of spectral imaging, a spectral filter array is much more difficult to produce than the color arrays used in commercial color cameras. Color cameras generally employ pigment-based filters whose spectral transmission bandwidth is too wide and irregular for the radiometric measurements of spectral imaging. The number of different filters that can be produced in a pigment-based color filter array is also limited by the number of pigments used. The cost of a color filter array thus greatly increases with the increase in the number of different filters.
Interferometric filters, such as Fabry-Perot filters, are based on a different principle than pigment filters and enjoy a number of properties useful for imaging spectroscopy:                finely tunable central wavelength: the central wavelength is given by the distance between the reflective surfaces used to make light interfere with itself. Varying this distance changes the central wavelength of the filter without significantly altering the shape of its spectral transmission curve;        symmetry around central wavelength: the filter transmission is locally symmetric with respect to the central wavelength, due to the natural interferometric process;        finely tunable bandwidth: the filter bandwidth is given by the reflectivity of the surfaces which reflect the light;        the production cost of a large number of different wavelength interferometric filters is much lower than for pigment-based filters.        
While their properties make them an obvious choice for imaging spectroscopy, until recently, interferometric filters were not feasible to create at the size of a single pixel. Fabry-Perot filters were thus used to produce spectral images by band-sequential scanning, as large global filters, which cover the entire sensor surface [8]. However, recent developments in miniaturization and nanotechnology have led to the development of interferometric spectral filter arrays with the individual filters being no larger than a pixel [4].
One of the main disadvantages of interferometric filters is their second and higher order transmissions. These higher order transmissions of the filters let light pass not only at the desired central wavelength, but also at twice that wavelength, three times that wavelength and so on. Interferometric filters can also be configured as multiband filters, due to the higher order transmissions which include multiple specific wavelengths for which they transmit light. In most cases, the higher order transmissions are undesired effects and need to be nullified.